Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,9,10,6)(4,7,14,13)(11,16,12,15)(17,18)(19,24)(20,22)(21,28)(23,27)(25,29)(26,30)(31,32)(33,34,38,41)(35,39,44,37)(36,40,46,48)(42,47,43,45);(3,10)(6,9)(11,12)(15,16)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,35,38,44)(34,39,41,37)(36,43,46,42)(40,45,48,47);(1,3)(2,6)(4,11)(5,10)(7,15)(8,9)(12,14)(13,16)(17,18,22,20)(19,24,27,23)(21,25,29,28)(26,31,32,30)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44);(1,3)(2,6)(4,12)(5,10)(7,16)(8,9)(11,14)(13,15)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,34)(35,37)(36,40)(38,41)(39,44)(42,45)(43,47)(46,48);(1,4,5,14)(2,7,8,13)(3,11,10,12)(6,15,9,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,36)(34,40)(35,43)(37,47)(38,46)(39,45)(41,48)(42,44);(1,2)(3,6)(4,13)(5,8)(7,14)(9,10)(11,16)(12,15)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,37)(34,35)(36,45)(38,39)(40,42)(41,44)(43,48)(46,47)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10371774);